" So, why did they have summer and winter soap?..."
From my reading, it's because the fats in a summer recipe gave a firmer soap, so the summer soap would not be as runny in the summer heat. The opposite would be true in winter -- you could make a softer type of soft soap and still have it be not-runny because your home would be a lot colder. The quote below is about soft (potassium) soap recipes. I know some hard (sodium) soap makers also had summer and winter formulas, but I haven't been able to find the source to share with you.
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"...The fabrication of soft soaps will now be described. Soft soap is a more or less impure solution of potash soap mixed with glycerin in caustic lye, and forming at ordinary temperatures a transparent smeary jelly, containing at times, and especially in cold weather, white grains, which are impure potassium or sodium stearates....
"In England, whale, seal and linseed oils are chiefly used, and occasionally a little tallow to produce the grains, or figging just described, an appearance which serves no really useful purpose. On the European continent, hemp-seed, linseed, camelina, and poppy oils are used...
[R]apeseed and train oils [are also used], especially in summer, since they produce a harder soap. In America, cottonseed oil and oleic acid are often employed...."
Source: J. Veitch Wilson & William Lant Carpenter. A Treatise on the Manufacture of Soap and Candles, Lubricants and Glycerin. 1895.
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It's interesting to me that the "stearic spots" we all love to hate nowadays were a desirable feature called "figging" in the soft soaps of the 1800s.
"Train oil" is oil rendered from whale blubber. According to Wikipedia, the word train in this context "...comes from the Dutch word traan ("tear" or "drop")...."
https://en.wikipedia.org/wiki/Whale_oil